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@ -20,7 +20,7 @@
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= Bonus problems
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#problem()
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Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ iff $gcd(x, n) = 1$
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Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ if and only if $gcd(x, n) = 1$
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#v(1fr)
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@ -45,7 +45,7 @@ How many permutations of $n$ objects are there?
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#v(1fr)
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#problem()
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What map corresponds to the permutation that produces the array `312` from the array `123`?
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What map corresponds to the permutation that produces the array `231` from the array `123`?
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#v(1fr)
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@ -100,7 +100,7 @@ How about $[321][213][231]$? \
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Rewrite these compositions as one permutation in square brackets.
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#solution([
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- $[1324][4321]$ is $[4321]$
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- $[1324][4321]$ is $[4231]$
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- $[321][213][231]$ is $[123]$
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])
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@ -501,7 +501,7 @@ List all other ways to write this cycle. \
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#definition("Inverse")
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The _inverse_ of a permitation $f$ is a permutation $g$ that "un-does" $f$. \
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The _inverse_ of a permutation $f$ is a permutation $g$ that "un-does" $f$. \
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This means that $g(f(x)) = x$ for all $x$.
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#problem()
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@ -550,7 +550,7 @@ Show that any cycle $(123...n)$ is equal to the product $(12)(23)...(n-1, n)$.
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Write $(7126453)$ as a product of transpositions. \
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#solution[
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Move elements one at a time, and using the last position as temporary storage.
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Move elements one at a time, using the last position as temporary storage.
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We get $(71)(72)(76)(74)(75)(73)$.
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Other solutions are possible. \
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@ -589,7 +589,7 @@ Show that any permutation is a product of transpositions of the form $(1, k)$. \
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#problem(label: "oneplustrans")
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Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$.
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Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$ whenever $a + 1 != b$.
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#solution[
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This is the same as @onetrans,
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@ -2,20 +2,16 @@
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#import "@preview/cetz:0.4.2"
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#import "../macros.typ": *
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= Groups (review)
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= Groups
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#definition()
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Before we continue, we must introduce a bit of notation:
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- $S_n$ is the set of permutations on $n$ objects.
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- $ZZ_n$ is the set of integers mod $n$.
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- $ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \
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In other words, it is the set of integers smaller than $n$ and coprime to $n$.#footnote[We proved this in another handout, but you may take it as fact here.] \
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For example, $ZZ_12^times = {1, 5, 7, 11}$.
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#problem()
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What are the elements of $S_3$? #hint[Use cycle notation] \
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How about $ZZ_17^times$?
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How about $ZZ_8$?
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#v(1fr)
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@ -47,6 +43,16 @@ What is the group with the fewest number of elements?
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Verifying that the trivial group is a group is trivial.
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]
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#definition()
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$ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \
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We can prove that this is the set of integers smaller than $n$ and coprime to $n$. \
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For example, $ZZ_12^times = {1, 5, 7, 11}$.
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#problem()
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What are the elements of $ZZ^times_8$? \
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How about $ZZ^times_23$? #hint[23 is prime.]
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#v(1fr)
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#pagebreak()
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@ -66,8 +72,11 @@ Show that $S_n$ is a group under composition.
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#problem()
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Let $(G, *)$ be a group with finitely many elements, and let $a in G$. \
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Show that there is an $n$ in $in ZZ^+$ so that $a^n = e$ \
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#hint[$a^n = a * a * ... * a$ repeated $n$ times.]
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Show that there is an $n$ in $in ZZ$ so that $a^n = e$ \
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#hint[
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$a^n = a * a * ... * a$ repeated $n$ times. \
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$a^(-n) = a^(-1) * a^(-1) * ... * a^(-1)$, where $a^(-1)$ is the inverse of $a$. \
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]
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#v(2mm)
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@ -111,7 +120,7 @@ Then, find all generators of $(ZZ_5, +)$
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How many groups have only one generator?
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#solution[
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Only one: the trivial group. The inverse of a generator is also a generator!
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Two: the trivial group and $(ZZ_2, +)$.
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]
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#v(1fr)
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